Question

A necessary and sufficient condition that the general equation of second degree \(ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0\) may represent a pair of straight lines is

• A. \( abc + 2fgh - af^2 - bg^2 - ch^2>0 \)
• B. \( abc + 2fgh - af^2 - bg^2 - ch^2<0 \)
• C. \( abc + 2fgh - af^2 - bg^2 - ch^2 = 0 \)
• D. \( abc + 2fgh - af^2 - bg^2 - ch^2 = a^2 + b^2 + c^2 \)

Hint:

A useful mnemonic to remember the expression for the determinant is "A Hot Girl Had Beautiful Face, Go For Chocolates":
\(a(bc - f^2)\)
\(-h(hc - fg)\)
\(+g(hf - bg)\)
Another way is to remember the phrase: "All handsome guys having beautiful faces go for coffee", which helps to set up the matrix \(\begin{pmatrix} a & h & g
h & b & f
g & f & c \end{pmatrix}\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The general second-degree equation represents a conic section. Under a specific condition on its coefficients, this conic section degenerates into a pair of straight lines. This condition can be expressed using the determinant of a related matrix.
Step 2: Key Formula or Approach:
The general second-degree equation can be represented in matrix form using homogeneous coordinates. The condition for the equation to represent a pair of straight lines is that the determinant of the associated 3x3 symmetric matrix is zero.
The matrix is: \[ \Delta = \begin{vmatrix} a & h & g
h & b & f
g & f & c \end{vmatrix} \] The condition is \( \Delta = 0 \).
Step 3: Detailed Explanation:
We need to compute the determinant of the matrix \( \Delta \). \[ \det(\Delta) = a(bc - f^2) - h(hc - fg) + g(hf - bg) \] \[ = abc - af^2 - h^2c + fgh + fgh - bg^2 \] \[ = abc + 2fgh - af^2 - bg^2 - ch^2 \] For the equation to represent a pair of straight lines, this determinant must be zero. \[ abc + 2fgh - af^2 - bg^2 - ch^2 = 0 \] Step 4: Final Answer:
The necessary and sufficient condition for the general equation of the second degree to represent a pair of straight lines is \( abc + 2fgh - af^2 - bg^2 - ch^2 = 0 \).